Robust Optimization:Applications in

Portfolio Selection Problems

Vris Cheung and Henry Wolkowicz

WatRISQ

University of Waterloo

Vris Cheung (University of Waterloo) Robust optimization 2009 1 / 19

Outline

◦ Motivations

◦ Robust optimization: modeling and computationalefficiency

◦ Applications

• Mean-variance model

• Sharpe ratio maximization

Vris Cheung (University of Waterloo) Robust optimization 2009 2 / 19

Motivations

Motivations

Problem input data frequently suffers from estimation error, andsometimes even very small errors can render the “optimal” solutionirrelevant (e.g. seriously infeasible).

e.g. Pilot 4 from NETLIB : constraint 372

aTx = − 15.79081x826 − 8.598819x827 − 1.88789x828 − 1.362417x829

− 1.526049x830 − 0.031883x849 − 28.725555x850 − 10.792065x851

− 0.19004x852 − 2.757176x853 − 12.290832x854 + 717.562256x855

− 0.057865x856 − 3.785417x857 − 78.30661x858 − 122.163055x859

− 6.46609x860 − 0.48371x861 − 0.615264x862 − 1.353783x863

− 84.644257x864 − 122.459045x865 − 43.15593x866 − 1.712592x870

− 0.401597x871 + x880 − 0.946049x898 − 0.946049x916

� b = 23.387405

Vris Cheung (University of Waterloo) Robust optimization 2009 3 / 19

Motivations

Motivations

Problem input data frequently suffers from estimation error, and because ofthis the “optimal” solution computed is far from optimal in reality.

e.g. Computation of efficient frontier from mean-variance model

◦ True asset returns μ and covariance matrix Q

◦ Estimate asset returns μ and covariance matrix Q

maxx

μTx − λxTQx s.t. 1Tx = 1 (QPλ)

Family of true optimal portfolio {xλ : xλ solves (QPλ), λ > 0}True efficient frontier : {(

√xTλQxT

λ,μTxλ) : λ > 0}(↑ What you would have obtained if you knew the true parameters μ and Q.)

Vris Cheung (University of Waterloo) Robust optimization 2009 4 / 19

Motivations

Motivations

Problem input data frequently suffers from estimation error, and because ofthis the “optimal” solution computed is far from optimal in reality.

e.g. Computation of efficient frontier from mean-variance model

◦ True asset returns μ and covariance matrix Q

◦ Estimate asset returns μ and covariance matrix Q

maxx

μTx − λxTQx s.t. 1Tx = 1 (QPλ)

Family of “optimal” portfolio {xλ : xλ solves (QPλ), λ > 0}(↑ What you actually obtain based on the estimates μ and Q.)

Estimated efficient frontier : {(√

xTλQxT

λ, μTxλ) : λ > 0}(This efficient frontier is what you think you would get.)

Vris Cheung (University of Waterloo) Robust optimization 2009 4 / 19

Motivations

Motivations

Problem input data frequently suffers from estimation error, and because ofthis the “optimal” solution computed is far from optimal in reality.

e.g. Computation of efficient frontier from mean-variance model

◦ True asset returns μ and covariance matrix Q

◦ Estimate asset returns μ and covariance matrix Q

maxx

μTx − λxTQx s.t. 1Tx = 1 (QPλ)

Family of “optimal” portfolio {xλ : xλ solves (QPλ), λ > 0}(↑ What you actually obtain based on the estimates μ and Q.)

Actual efficient frontier : {(√

xTλQxT

λ,μTxλ) : λ > 0}(This efficient frontier is what you actually get in reality!)

Vris Cheung (University of Waterloo) Robust optimization 2009 4 / 19

Motivations

Motivations

Source: Computing Efficient Frontiers using Estimated Parameters, M. Broadie, 1993, Annals of Operations Research, Vol.

45, 21-58.

Vris Cheung (University of Waterloo) Robust optimization 2009 5 / 19

Motivations

Motivations

◦ Problem input data frequently suffers from estimationerror.

◦ Solving an optimization problem based on nominal dataalone could produce some “optimal solution” that isirrelevant in reality.

◦ Even if feasibility is not an issue, the “optimal” solutionobtained is very likely far from optimal.

Vris Cheung (University of Waterloo) Robust optimization 2009 6 / 19

Robust optimization and efficiency

Robust optimization: modeling

◦ Nominal LP problem:

minx

cTx + d s.t. Ax � b

◦ Uncertain data (A, b) takes value in UA,b (and (c, d) in Uc,d)

◦ Robust counterpart:

minx

c(x) s.t. Ax � b for all (A, b) ∈ UA, b

where c(x) := sup(c, d)∈Uc, d

cTx + d .

Vris Cheung (University of Waterloo) Robust optimization 2009 7 / 19

Robust optimization and efficiency

Robust optimization: modeling

◦ Nominal LP problem:

minx

cTx + d s.t. Ax � b

◦ Uncertain data (A, b) takes value in UA,b (and (c, d) in Uc,d)

◦ Robust counterpart:

minx

c(x) s.t. Ax � b for all (A, b) ∈ UA, b

where c(x) := sup(c, d)∈Uc, d

cTx + d .

Vris Cheung (University of Waterloo) Robust optimization 2009 7 / 19

Robust optimization and efficiency

Robust optimization: modeling

◦ Nominal LP problem:

minx

cTx + d s.t. Ax � b

◦ Uncertain data (A, b) takes value in UA,b (and (c, d) in Uc,d)

◦ Robust counterpart:

minx

c(x) s.t. Ax � b for all (A, b) ∈ UA, b

where c(x) := sup(c, d)∈Uc, d

cTx + d .

Vris Cheung (University of Waterloo) Robust optimization 2009 7 / 19

Robust optimization and efficiency

Robust optimization: modeling

◦ Nominal problem:

minx

f (x; η) s.t. g(x; ξ) � 0

◦ Uncertain data η and ξ vary withinUη and Uξ resp.

◦ Robust counterpart:

minx

f (x) s.t. g(x;ξ) � 0 for all ξ ∈ Uξ

where f (x) := supη∈Uη

f (x;η).

Vris Cheung (University of Waterloo) Robust optimization 2009 8 / 19

Robust optimization and efficiency

Robust optimization: modeling

◦ But what are the uncertainty sets?

◦ In finance, the uncertainty structure corresponds to theconfidence region.

◦ So this problem is deterministic, but is a semi-infinite(and possibly non-smooth) programming problem!

◦ Infinite number of constraints poses computationaldifficulty.

◦ But constraints such as the last one can be dealt withefficiently.

Vris Cheung (University of Waterloo) Robust optimization 2009 9 / 19

Robust optimization and efficiency

Robust optimization: efficiency

An illustration in LP case:

◦ Consider an uncertain constraint

aTx � b where [a; b] ={[a0; b0] +

L∑l=1

ζl[al; bl] : ‖ζ‖2 � 1

}

◦ Robust counterpart:

(a0)Tx +

L∑l=1

ζl[al]Tx � b0 +

L∑l=1

ζlbl ∀ ‖ζ‖2 � 1

Vris Cheung (University of Waterloo) Robust optimization 2009 10 / 19

Robust optimization and efficiency

Robust optimization: efficiency

An illustration in LP case:

◦ Consider an uncertain constraint

aTx � b where [a; b] ={[a0; b0] +

L∑l=1

ζl[al; bl] : ‖ζ‖2 � 1}

◦ Robust counterpart:∥∥([a1]Tx − b1, . . . , [aL]Tx − bL)T

∥∥2 � b0 − (a0)Tx

which is not a “hard” constraint to deal with.

Vris Cheung (University of Waterloo) Robust optimization 2009 10 / 19

Robust optimization and efficiency

Robust optimization: efficiency

Robust counterparts of some classes of non-linearprogramming problems and LP with different uncertainty setsmay have a finite representation, and can be solved efficiently.

Vris Cheung (University of Waterloo) Robust optimization 2009 11 / 19

Applications

Mean-variance model

(λ-)parametric QP

maxx

μTx − λxTQx s.t. 1Tx = 1

where (assuming n stocks are available to choose)

x ∈ Rn : proportion of investment on the available assets

μ ∈ Rn : expected return of the available assets

Q ∈ Rn×n : covariance matrix of the available assets

λ > 0 : risk aversion parameter

Vris Cheung (University of Waterloo) Robust optimization 2009 12 / 19

Applications

Mean-variance model

(μ0-)parametric QP

minx

xTQx s.t. μTx � μ0 , 1Tx = 1

where (assuming n stocks are available to choose)

x ∈ Rn : proportion of investment on the available assets

μ ∈ Rn : expected return of the available assets

Q ∈ Rn×n : covariance matrix of the available assets

μ0 ∈ R : minimum expected return guarantee

Vris Cheung (University of Waterloo) Robust optimization 2009 12 / 19

Applications

Mean-variance model

Uncertain (μ0-)parametric QP

minx

xTQx s.t. μTx � μ0 , 1Tx = 1

Robust counterpart

minx

{maxQ∈UQ

xTQx : μTx � μ0 ∀μ ∈ Uμ , 1Tx = 1}

UQ : uncertainty set for Q

Uμ : uncertainty set for μ

Vris Cheung (University of Waterloo) Robust optimization 2009 13 / 19

Applications

Mean-variance model

Robust counterpart:

minx

{maxQ∈UQ

xTQx : μTx � μ0 ∀μ ∈ Uμ , 1Tx = 1}

Some special cases:

◦ Q is certain :

Uμ = {μ : μ− δ � μ � μ + δ}

=⇒ minx

xTQx s.t. μTx − δT|x| � μ0 , 1Tx = 1

Vris Cheung (University of Waterloo) Robust optimization 2009 14 / 19

Applications

Mean-variance model

Robust counterpart:

minx

{maxQ∈UQ

xTQx : μTx � μ0 ∀μ ∈ Uμ , 1Tx = 1}

Some special cases:

◦ Q is certain :

Uμ = {μ : (μ − μ)TQ−1(μ − μ) � χ2}

=⇒ minx

xTQx s.t. μTx � μ′0 , 1Tx = 1

for some μ ′0 � μ0.

Vris Cheung (University of Waterloo) Robust optimization 2009 14 / 19

Applications

Mean-variance model

0 5 10 15 20 250

2

4

6

8

10

12

14

16

18(σ, μ) frontiers realized with nominal and worst−case inputs from 4−year moving averages

Standard deviation of efficient portfolios (annualized percentages)

Exp

ecte

d re

turn

of e

ffici

ent p

ortfo

lios

(ann

ualiz

ed p

erce

ntag

es)

Classical efficient portfoliosRobust efficient portfolios (4 yr.)Robust efficient portfolios (2 yr.)

Source: Robust Asset Allocation, R.H. Tutuncu and M. Koenig. Annals of Operations Research, 132, 2004.

Vris Cheung (University of Waterloo) Robust optimization 2009 15 / 19

Applications

Sharpe ratio maximization

(σm

, μm

)

(σp , μ

p)

rf

0

Portfolio return

Portfolio SD

maxx

μTx − rf√xTQx

s.t. 1Tx = 1

(rf = risk free rate)

Vris Cheung (University of Waterloo) Robust optimization 2009 16 / 19

Applications

Sharpe ratio maximization

Nominal problem:

maxx

μTx − rf√xTQx

s.t. 1Tx = 1 , x � 0

Robust counterpart:

maxx

{minμ, Q

μTx − rf√xTQx

s.t. μ ∈ Uμ , Q ∈ UQ

}s.t. 1Tx = 1 , x � 0

It can be shown that the above max-min problem can bereduced to a SOCP (with 2n + 6 variables and 8 constraints).

Vris Cheung (University of Waterloo) Robust optimization 2009 17 / 19

Applications

Sharpe ratio maximization

Advantages of robustification:

◦ Lower turnover [Ceira]• At 95% confidence level, turnover drops by 4%.• At 99% confidence level, turnover drops by 7%.

◦ This indicates a lower aggregate transaction cost.

◦ Higher terminal wealth [Goldfarb and Iyengar]• At 95% confidence level, final wealth is 40% higher.• At 99% confidence level, final wealth is 50% higher.

Vris Cheung (University of Waterloo) Robust optimization 2009 18 / 19

Applications

Summary

◦ Robust optimization can give uncertainty-immunesolutions

◦ Many robust optimization problems can be solvedefficiently

◦ Robust optimization can producing “stable” solutions=⇒ lower turnover rate=⇒ suitable for long term planning

Vris Cheung (University of Waterloo) Robust optimization 2009 19 / 19